Runtime Analysis of a Simple Multi-Objective Evolutionary Algorithm

Practical knowledge on the design and application of multiobjective evolutionary algorithms (MOEAs) is available but well-founded theoretical analyses of the runtime are rare. Laumanns, Thiele, Zitzler, Welzel and Deb (2002) have started such an analysis for two simple mutation-based algorithms (SEMO and FEMO) for combinatorial optimization problems. These algorithms search locally in the neighborhood of their current population by selecting an individual and flipping one randomly chosen bit. Due to their local search operator they cannot escape from local optima, and, therefore, they have no finite expected runtime in general. We investigate the runtime of a variant of SEMO whose mutation operator flips each bit independently. It is proven that its expected runtime is O(n) for all objective functions f : {0, 1} → R, i. e., independently of the number of objectives m. There are bicriteria problems among the hardest problems for this algorithm. Moreover, for each d between 2 and n, a bicriteria problem with expected runtime Θ(n) is presented. This shows that bicriteria problems cover the full range of potential runtimes of this variant of SEMO. For the problem LOTZ (leading ones trailing zeroes), the runtime does not increase substantially if we use the global search operator. Finally, we consider the problem MOCO (multi-objective counting ones). We show that the conjectured bound O(n logn) on the expected runtime is wrong for both variants of SEMO. In fact, MOCO is almost a worst case example for SEMO if we consider the expected runtime; however, the runtime is O(n logn) with high probability.